Rational points on a subanalytic surface

Jonathan Pila[1]

  • [1] McGill University, department of mathematics and statistics, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, H3A 2K6 (Canada), University of Oxford, mathematical institute, 24-29 St Giles, Oxford OX1 3LB (Grande-Bretagne)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 5, page 1501-1516
  • ISSN: 0373-0956

Abstract

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Let X ⊂ ℝ n be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of X that do not lie on some connected semialgebraic curve contained in X .

How to cite

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Pila, Jonathan. "Rational points on a subanalytic surface." Annales de l’institut Fourier 55.5 (2005): 1501-1516. <http://eudml.org/doc/116224>.

@article{Pila2005,
abstract = {Let $X\subset \{\mathbb \{R\}\}^n$ be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of $X$ that do not lie on some connected semialgebraic curve contained in $X$.},
affiliation = {McGill University, department of mathematics and statistics, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, H3A 2K6 (Canada), University of Oxford, mathematical institute, 24-29 St Giles, Oxford OX1 3LB (Grande-Bretagne)},
author = {Pila, Jonathan},
journal = {Annales de l’institut Fourier},
keywords = {Subanalytic set; rational point; subanalytic set; semianalytic set; height},
language = {eng},
number = {5},
pages = {1501-1516},
publisher = {Association des Annales de l'Institut Fourier},
title = {Rational points on a subanalytic surface},
url = {http://eudml.org/doc/116224},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Pila, Jonathan
TI - Rational points on a subanalytic surface
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 5
SP - 1501
EP - 1516
AB - Let $X\subset {\mathbb {R}}^n$ be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of $X$ that do not lie on some connected semialgebraic curve contained in $X$.
LA - eng
KW - Subanalytic set; rational point; subanalytic set; semianalytic set; height
UR - http://eudml.org/doc/116224
ER -

References

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  1. E. Bierstone, P.D. Milman, Semianalytic and subanalytic sets, Pub. Math. I.H.E.S 67 (1988), 5-42 Zbl0674.32002MR972342
  2. E. Bombieri, email, 9 March, (2003) 
  3. E. Bombieri, J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337-357 Zbl0718.11048MR1016893
  4. D. R. Heath-Brown, The density of rational points on curves and surfaces, Ann. Math. 155 (2002), 553-595 Zbl1039.11044MR1906595
  5. M. Hindry, J. H. Silverman, Diophantine geometry: an introduction, 201 (2000), Springer, New York Zbl0948.11023MR1745599
  6. V. Jarnik, Über die Gitterpunkte auf konvexen Curven, Math. Z. 24 (1926), 500-518 MR1544776
  7. S. Lang, Number theory III: diophantine geometry, 60 (1991), Springer, Berlin Zbl0744.14012MR1112552
  8. J. Pila, Geometric postulation of a smooth function and the number of rational points, Duke Math. J. 63 (1991), 449-463 Zbl0763.11025MR1115117
  9. J. Pila, Density of integer points on plane algebraic curves, International Mathematics Research Notices (1996), 903-912 Zbl0973.11085MR1420555
  10. J. Pila, Integer points on the dilation of a subanalytic surface, Quart. J. Math. 55 (2004), 207-223 Zbl1111.32004MR2068319
  11. J. Pila, Note on the rational points of a pfaff curve Zbl1097.11037
  12. W. M. Schmidt, Integer points on curves and surfaces, Monatsh. Math. 99 (1985), 45-72 Zbl0551.10026MR778171
  13. H. P. F. Swinnerton-Dyer, The number of lattice points on a convex curve, J. Number Theory 6 (1974), 128-135 Zbl0285.10020MR337857
  14. A. Wilkie, Diophantine properties of sets definable in o -minimal structures, J. Symb. Logic 69 (2004), 851-861 Zbl1081.03038MR2078926

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